- #1

PWiz

- 695

- 116

In Euclidean space, $$ds^2= dx^2+dy^2+dz^2$$

Where ##x=r \ cos\theta \ sin\phi## , ##y=r \ sin\theta \ sin\phi## , ##z=r \ cos\phi## (I'm using ##\phi## for the polar angle)

For simplicity, let ##cos \phi = A## and ##sin \phi = B##

Then ##dx=r \ cos\theta \ dB + B(cos \theta \ dr - r \ sin \theta \ d \theta)##

##dy=r \ sin\theta \ dB + B(sin \theta \ dr + r \ cos \theta \ d \theta)##

##dz= r \ dA + A \ dr##

When I add up the squares of the 3 differential elements above, gather similar terms and replace ##A## and ##B##, I get ##ds^2 = dr^2 + r^2 \ sin^2 \phi \ d \theta^2## , and you can see that one term is missing. Is something wrong with my approach?